Study On A Class Of Time-dependent Quasivariational-Hemivariational Inequality Problems

With the development of nonlinear analysis and other disciplines,hemivariational inequalities,as an extension of variational inequalities,have important applications in the fields of contact mechanics,fluid mechanics and other fields.In recent years,an important class of hemivariational inequalities named quasi-hemivariational inequalities,characterized by the correlation between constraint sets and solutions,has attracted the attention of many scholars.Compared to general hemivariational inequalities,the research of quasihemivariational inequalities is relatively difficult,which leads to fewer research results on this problem at present.Therefore,the theory and application of quasi-hemivariational inequality problem have become a hot topic in current research.In this thesis,a class of time-dependent quasi-variational-hemivariational inequalities involving both convex and nonconvex,nonsmooth terms is considered,and the constraint set is related with solutions.By using the principle of measurability and other tools,the solvability and continuous dependence of this problem is studied.The specific research content is as follows:In the beginning,using the existing results in the literature,a set-valued mapping is defined by the solvability of an auxiliary problem which is a class of quasi-variationalhemivariational inequalities.By using the measurable selection lemma,it is obtained that there exists a measurable selection of this set-valued mapping,and the measurable selection is proved to be the solution to the time-dependent quasi-variational–hemivariational inequality problem studied in the quadratic integrable function space.After that,by enhancing the conditions of the constraint set,the uniqueness of the solutions to the original problem is obtained.Further,on the basis that the original problem has a unique solution,the perturbation problem of the original problem is considered,it is proved that any solution sequence of the perturbation problem strongly converges to the unique solution of the original problem in the quadratic integrable function space.Finally,as an application,this thesis considers a frictional contact problem with locking materials.By virtue of Green’s formula,the variational-hemivariational model for the frictional contact problem is drawn,and the existence,uniqueness and stability of the weak solutions to the contact problem are derived by using the above theoretical results.The class of time-dependent quasi-variational-hemivariational inequalities studied above is a generalization of the hemivariational inequality in the literature.The theoretical results obtained in this thesis not only generalize the relevant conclusions in the literature,but also can be applied to frictional contact problems.In conclusion,the research of this problem has theoretical and practical value.